Econ 3710 - Youngstown State University



Practice Problem Answers. Study hard and you will do well! See you Tuesday 5/5 at 3:15 pm.

1. If the marginal product of labor is currently 10, and the marginal rate of technical substitution is currently 20, what does the marginal product of capital equal?

MRTS=MPL/MPK; 20=10/MPK; MPK =0.5

2. A firm has access to two production processes with the following marginal cost curves:

MC1=0.4Q and MC2= 2 + 0.2Q

If it wants to produce 8 units of output, how much should it produce with each process?

The firm minimizes costs when it distributes production across the two processes so that marginal cost is the same in each. If Q1 denotes production in the first process and Q2 is production in the second process, we have Q1 + Q2 = 8 and 0.4Q1 = 2 + 0.2Q2, which yields Q1=6, Q2=2. The common value of marginal cost will be 2.4.

3. You have been hired by the IRS to audit a dead economist's consulting firm. He left his records incomplete so that only a student trained in economics could fill in the blanks. . You have been hired by the IRS to audit a dead economist's consulting firm. He left his records incomplete so that only a student trained in economics could fill in the blanks.

Clients |Total Cost |Total Fixed Cost |Total Variable Cost |Average Total Cost |Average Variable Cost |Average Fixed Cost |Marginal Cost | |0 |100 |100 |0 |- |- |- | | |1 |175 |100 |75 |175 |75 |100 |75 | |2 |240 |100 |140 |120 |70 |50 |65 | |3 |330 |100 |230 |110 |83.33 |33.33 |90 | |4 |450 |100 |350 |112.5 |87.5 |25 |120 | |5 |600 |100 |500 |120 |100 |20 |150 | |6 |800 |100 |700 |133.33 |116.67 |16.67 |200 | |7 |1100 |100 |1000 |157.14 |142.86 |14.29 |300 | |Does the consulting firm exhibit diminishing returns for any number of clients between 0 and 7?

The firm exhibits diminishing returns for any number of clients greater than two.

4. Doug wants to go into the donut business. For $500 per month he can rent a bakery complete with all the equipment he needs to make a dozen different kinds of donuts (K=l). He must pay unionized donut bakers a monthly salary of $400 each. He projects his production function to be Q = 5KL (where Q is tons of donuts).

a. What is Doug's monthly total cost function, variable cost function, and marginal cost?

b. How many bakers will Doug hire to make 25 tons of donuts?

c. What will happen to Doug's total cost if his production function turns out to be Q=2KL?

a) K=1 L=Q/5

TC = 500 + 400(Q/5) = 500 + 80Q

VC = 80Q

The marginal cost is equal to the slope of the total cost curve which is MC = 80.

b) L = Q/5 = 25/5 = 5

c) K = 1, L = Q/2, TC = 500 + (400/2)Q = 500 + 200Q

5. A firm’s production is given by Q = K1/3L2/3; the price of labor (w) equals $12 and the price of capital (r) equals $16.

a.) What is the optimal amount of capital and labor needed to produce 2,000 units?

b.) What do their total costs equal?

a. The MRTS=(2K/L) = (w/r=12/16); L=2.67K or K=.375L. Plugging this back into the production function one finds:

2000 = (.375K)1/3L2/3=.7214L

L*=2772; K*=1040

b. TC = 2772*$12 + 1040*16 = $49903 (rounded)

6. Suppose your firm uses labor and capital in the production of commodity X. You are currently employing 15 workers and 30 units of capital. The total wage bill (L x w) is $800 per day, and the total capital payments bill is $1200 per day (K x r).

a. Suppose that you know that the marginal physical products of labor and capital are 20 and 18 respectively. Is your firm currently minimizing long run costs? Explain.

b. Alternatively, assume that your firm is indeed minimizing long run costs. What is the long run marginal cost of output if you are certain that the marginal product of capital, MPk, at the current output rate is 10? Explain.

a. No. After you translate the total wage bills into prices per unit of input, the current mix of inputs does not meet the cost-minimizing condition: ratio of marginal products must equal the ratio of input prices, or marginal product per dollar must be equal across all inputs.

b. If you are currently minimizing costs, you are using inputs so that the marginal products per dollar are equal across inputs. Thus, with the rental price equal to $40, and MPk=10,

MC=r/MPk=$4.00

7. Assume a firm has total revenue function given by TR = 80Q and a total cost function given by TC = 200 + Q2. Solve for their profit maximizing quantity and price. What do their profits equal?

Profit maximizing Q is found by solving for:

MR = MC

80 = 2Q

Q* = 40

Profits = TR – TC = 80*40 – (200 + 402 ) = 3200 – 1800 = 1400

8. Suppose a representative firm in a perfectly competitive, constant cost industry has a cost function TC = 4Q2 + 100Q + 100.

a. What is the long-run equilibrium price for this industry?

The minimum point on LAC is found either by graphing the LAC curve or by taking the first

derivative and setting it to zero.

dLAC/dQ = 4 - 100/Q2 = 0, which yields Q = 5.

In the long run P = LAC = 140

b. If market demand is given by the function Q = 1000 – P, where P denotes price, what is the total quantity produce sold in this market? If all firms produce the same quantity, what is the total number of firms in this market?

If demand is Q=1000-P, then at P=140, we get Q=860. So in long run equilibrium, there will be

860/5 = 172 firms.

9. Use the following production/cost table to answer the following questions.

Labor Output(Q) Fixed Costs Variable Costs Total Costs Marginal Costs

0 0 400 0 400

1 10 400 50 450 50/10=$5

2 22 400 100 500 50/12=$4.17

3 30 400 150 550 50/8=$6.25

4 38 400 200 600 50/8=$6.25

5 40 400 250 650 50/2=$25

6 41 400 300 700 50/1=$50

a. Fill in the blanks in the table above.

b. If this firm sells all its output at a constant price of $25, what is the profit maximizing Q? P=MC at Q=40.

c. What do their profits equal?

TR=P*Q – TC = $25*40 - $650 = $1,000 - $650 = $350

10. Use the following diagram representing a perfectly competitive firm to answer the following questions

a. What is the profit maximizing quantity? 600

b. What does profit equal at the profit maximizing Q? ($20 - $15)*600=3,000

c. What is the shutdown price? $6 (minimum of AVC)

d. At what price would profits equal zero? $14 (minimum of ATC)

e. If price changes from $20 to $14, what happens to this firm’s output? Output would decrease from 600 to 450.

[pic]

11. Refer to the graph above. If the market price increases from $50 per unit to $60 per unit, how will a profit-maximizing perfectly competitive firm change their output?

They will increase production from 65 to 75.

12. Suppose a firm is earning negative profit, but is continuing to operate. What do we know must be true of TC compared to TR, and VC compared to TR? What do we know must be true of ATC compared to P, and AVC compared to P?

Because profit is negative, we know that TRVC.

Similarly, we know that AVCMC

2 $9.00 $44000 4000 $9.00 $11.90 $10.74 a. MR=MC

3 $35.90 $37.90 5000 $37.90 $35.90 e. P ................
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